The missing ingredient is to prove that the weak flow constructed in Corollary REF  can be converted into an isotopy. In the setting of 2-convex mean curvature flow with surgery (see [1]}, [1]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}) this has been accomplished in several works [11]}, [12]}, [13]}, [14]}, [15]}. (We also mention related work using Ricci flow with surgery [16]}, [17]} and singular Ricci flow [18]}, [19]}, [20]}.) Given the resolution of the mean convex neighborhood conjecture [21]}, [22]}, it should be possible (although tedious) to adapt these results to the setting of singular mean curvature flows with only spherical and cylindrical singularities.
